MATHEMATICS OF COMPUTATION Volume 72, Number 242, Pages 913{937 S 0025-5718(02)01432-1 Article electronically published on February 15, 2002 CLASS NUMBERS OF REAL CYCLOTOMIC FIELDS OF PRIME CONDUCTOR RENE´ SCHOOF Abstract. + 1 The class numbers hl of the real cyclotomic fields Q(ζl + ζl− ) + are notoriously hard to compute. Indeed, the number hl is not known for a single prime l 71. In this paper we present a table of the orders of certain ≥ 1 subgroups of the class groups of the real cyclotomic fields Q(ζl + ζl− )forthe primes l<10; 000. It is quite likely that these subgroups are in fact equal to the class groups themselves, but there is at present no hope of proving this rigorously. In the last section of the paper we argue |on the basis of the Cohen-Lenstra heuristics| that the probability that our table is actually a + table of class numbers hl , is at least 98%. Introduction Let l>2 be a prime number and let ζl denote a primitive l-th root of unity. The ideal class group Cll of the ring of integers of the cyclotomic field Q(ζl)isa finite abelian group of order hl,theclass number of Q(ζl). The group Cll naturally splits into two parts; there is a natural exact sequence + 0 Cl Cll Cl− 0; −→ l −→ −→ l −→ + where Cll denotes the class group of the ring of integers of the real cyclotomic 1 1 + field Q(ζl + ζl− ). Its order, the class number of Q(ζl + ζl− ), is denoted by hl . The quotient group Cll− is rather well understood. Already in the 19th century, E.E. Kummer [12], [13] computed the orders of the groups Cll− for l<100. Nowa- days it is rather easy to compute these numbers for much larger values of l.See[19] for the structure of the groups Cll− and [5] for a study of the extension of Cll− by + + Cll . The present paper is concerned with the groups Cll . + The groups Cll are not well understood, and there is at present no practical method to compute their orders, not even for relatively small l. Methods that inspect all ideals of norm less than the classical Minkowski bound become useless 1 l 1 l 1 (l 1)=2 (l 3)=4 as l grows: for Q(ζl+ζl− ) the Minkowski bound is ( −2 )!( −2 )− − l − ,which exceeds 1028 when l>100. Algorithms that proceed by searching for fundamental units are not very efficient either, because the rank of the unit group of the ring 1 of integers of Q(ζl + ζl− )is(l 3)=2, which is at least 49 when l>100. This is too large to be of much use.− Using Odlyzko's discriminant bounds, F. van der + Linden computed in [22] the groups Cll for l 163. For l 71 his results are only valid under assumption of the Generalized Riemann≤ Hypothesis≥ for zeta functions of number fields. Van der Linden's results are the best known: strictly speaking, Received by the editor November 7, 2000 and, in revised form, July 9, 2001. 2000 Mathematics Subject Classification. Primary 11R18, 11Y40. c 2002 American Mathematical Society 913 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 914 RENE´ SCHOOF the largest prime l for which the class number of Q(ζl)isknownisl = 67. The class number of Q(ζ71) is unknown at present. Assuming the Generalized Riemann Hypothesis improves the situation only marginally: determining the class number of Q(ζ167) is beyond the scope of any known method. In view of this sorry state of affairs, we proceed in the following experimental 1 way. Let Gl denote the Galois group of Q(ζl +ζl− )overQ. This is a cyclic group of order (l 1)=2. Let Bl denote the quotient of the unit group of the ring of integers − 1 of Q(ζl + ζl− ) by its subgroup of cyclotomic units. It follows from the so-called class number formula [23] that + #Cll =#Bl: + This result can be refined as follows. Both groups Bl and Cll are finite Z[Gl]- modules and hence admit Jordan-H¨older filtrations with simple factors. An appli- 1 cation of the class number formula for Q(ζl + ζl− ) and its subfields shows that the + submodules of Bl and Cll all of whose simple Jordan-H¨older factors have some fixed order q, have the same number of elements as well. The simple Jordan-H¨older factors of the ring Z[Gl] are 1-dimensional vector spaces over the finite residue fields of Z[Gl]. On heuristic grounds one expects that when l varies, the smaller factors have a higher probability of occurring than the larger ones. Therefore we computed only the small Jordan-H¨older factors of Bl for primes l in a certain range. More precisely, our computation gives the following. Main result. A table of all the simple Jordan-H¨older factors of order less than 80; 000 of all groups Bl for l<10; 000. Moreover, we give their multiplicities and ~+ hence the order hl of the largest subgroup of Bl all of whose Jordan-H¨older factors ~+ have order less than 80; 000.Thenumberhl is also the order of the largest subgroup ~ + + Cll of Cll all of whose Jordan-H¨older factors have order less than 80; 000. This is a rather extensive calculation. We checked the more than 85 million simple Jordan-H¨older factors of the rings Z[Gl]withl<10; 000 that have order less than 80,000. Both bounds are rather arbitrary. It turned out that only 354 of the factors appear in the Jordan-H¨older filtration of the group Bl for some 1 field Q(ζl + ζl− ). The largest one has order 1451. For each occurring Jordan- H¨older factor we computed the multiplicity with which it occurs in Bl.More precisely, we determined for all l<10; 000 the Galois module structure of the largest submodule of Bl all of whose Jordan-H¨older factors have order less than ~ + 80; 000. This submodule is not necessarily isomorphic to the Galois module Cll , ~+ but it has the same order hl . ~+ + + ~+ We can say with certainty that hl divides hl and that either hl is equal to hl + ~+ ~+ + or that hl > 80; 000 hl . But we do not know for sure whether hl = hl forasingle l 71 (or l 167 if we· assume the Generalized Riemann Hypothesis). Nevertheless, an≥ informal≥ calculation based on the Cohen-Lenstra heuristics indicates that it is ~+ + not at all unlikely that actually hl = hl for all primes l in our range. Proving this rigorously seems completely out of reach however. + Note that we do not claim to have computed the p-part of hl for all primes ~+ p<80; 000. We show, for instance, that h167 = 1, but we have not even checked + that h167 = 1 is not divisible by 3! Indeed, since the relevant Jordan-H¨older factors 41 + 41 all have order 3 , once 3 divides h167,thensodoes3 . The heuristics indicate License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use CLASS NUMBERS OF REAL CYCLOTOMIC FIELDS OF PRIME CONDUCTOR 915 + that it is extremely improbable that Cl167 admits any simple Jordan-H¨older factors of order as large as 341. In section 1 we briefly discuss finite Gorenstein rings. Our main examples are finite group rings. In section 2 we give a description of the Galois modules Bl that is suitable for actual computation. In section 3 we explain how the calculations were performed. In section 4 we present the results of the calculations. The \Main ~+ Table" contains the numbers hl . In section 5 we show that, even if the Galois + modules Bl and Cll need not be isomorphic, their Galois cohomology groups are. For l<10; 000, they can readily be computed from the data given in section 4. Finally, in section 6, we present the heuristic arguments that lead to the assertion that with 98% probability, the Main Table is actually a table of class numbers of 1 Q(ζl + ζl− )forl<10; 000. Initially computations were performed on Macintoshes at the Universities of Sas- sari and Trento in Italy. PARI was used to do the multi-precision computations describedinsection3.IthankSt´ephane Fermigier, who some years later trans- lated my Pascal programs into C and ran them on a powerful Connection Machine in Paris, Larry Washington for several useful remarks, Francesco Pappalardi and Don Zagier for their help with the estimates in section 6, and Silvio Levy for the production of Figure 6.1. 1. Finite Gorenstein rings In this section we discuss some elementary properties of finite Gorenstein rings. The properties of these rings play a role in the next section. Let R be a finite commutative ring. For any R-module A, the additive group A? =HomR(A; R)isanR-module via (λf)(a)=λf(a)=f(λa)forλ R, dual 2 a A. Similarly, the dual group A =HomZ(A; Q=Z)isanR-module via (λf2)(a)=f(λa)forλ R, a A. The ring R is said2 to be 2Gorenstein if the R-module Rdual is free of rank 1 over R. For any positive M Z, the ring Z=M Z is a Gorenstein ring. If R is a finite Gorenstein ring, and g(X2 ) R[X] is a monic polynomial, then R[X]=(g(X)) is also a finite Gorenstein ring. In2 particular, for any M>0 and any finite abelian group G, the group ring (Z=M Z)[G] is Gorenstein.